# 08-401/Term Test

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## The Test

### Front Page

Math 401 Polynomial Equations and Fields

Term Test

University of Toronto, February 27, 2008

Solve 5 of the 6 problems on the other side of this page.

Each of the problems is worth 20 points.

You have two hours to write this test.

Notes.

• No outside material other than stationary and a basic calculator is allowed.
• The final exam date was posted by the faculty - it will take place on the evening of Tuesday April 28 between 7PM and 10PM, at BN2S (Large Gymnasium, South End, Benson Building, 320 Huron Street (south of Harbord Street), Second Floor).
• Neatness counts! Language counts! The \em ideal written solution to a problem looks like a proof from the textbook; neat and clean and made of complete and grammatical sentences. Definitely phrases like "there exists" or "for every" cannot be skipped. Lectures are mostly made of spoken words, and so the blackboard part of proofs given during lectures often omits or shortens key phrases. The ideal written solution to a problem does not do that.
• New! You will get 20% of the credit for any problem for which you will write explicitly "I don't know how to solve this problem" (whole problems only!).
Good Luck!

### Questions Page

Solve 5 of the following 6 problems. Each of the problems is worth 20 points. You have two hours. Neatness counts! Language counts!

Problem 1. Consider the ring ${\displaystyle \{0,2,4,6,8\}}$ under addition and multiplication modulo 10. Does it have a unity?

Tip. This, of course, is not just a yes/no question. You are expected to fully justify your answer, whatever it is.

Problem 2.

1. Define "an integral domain".
2. Define "a field".
3. Prove: A finite integral domain is a field.

Tip. As always in math exams, when proving a theorem you may freely assume anything that preceded it but you may not assume anything that followed it.

Problem 3. Let ${\displaystyle R}$ be a commutative ring with unity and let ${\displaystyle A}$ be an ideal in ${\displaystyle R}$. Prove that ${\displaystyle R/A}$ is a field if and only if ${\displaystyle A}$ is maximal.

Tip. Don't forget! There are two directions to prove here!

Problem 4. Find all ring homomorphisms from ${\displaystyle {\mathbb {Z} }/6}$ to ${\displaystyle {\mathbb {Z} }/10}$.

Tip. Here, of course, you have to explain both why the homomorphisms you found really are homomorphisms and why there are no more.

Problem 5.

1. Let ${\displaystyle F}$ be a field, let ${\displaystyle f}$ be a polynomial in ${\displaystyle F[x]}$, and let ${\displaystyle a}$ and ${\displaystyle b}$ be two different elements of ${\displaystyle F}$. Prove that the remainder for the division of ${\displaystyle f}$ by ${\displaystyle (x-a)(x-b)}$ is ${\displaystyle {\frac {f(b)-f(a)}{b-a}}(x-a)+f(a)}$.
2. Compute the remainder for the division exercise ${\displaystyle {\frac {x^{2008}}{x^{2}-1}}}$, done in ${\displaystyle {\mathbb {Q} }[x]}$.

Problem 6. Prove that the polynomial ${\displaystyle f=3x^{5}+15x^{4}-20x^{3}+10x+20}$ is irreducible over ${\displaystyle {\mathbb {Q} }}$. If you are using any major theorem, you need to quote it in full, but you don't need to prove it.

Good Luck!

## The Results

A total of 27 students took the exam; before appeals the average grade is 64.66 and the standard deviation is 18.79.

The full list of grades is: (31 43 43 46 47 49 50 52 53 54 55 60 60 61 61 62 65 73 75 75 76 82 83 92 99 99 100).

The results are quite similar to what I expected them to be, perhaps a bit on the low side.

• If you got 100 you should pat yourself on your shoulder and feel good.
• If you got something like 95, you're doing great. You made a few relatively minor mistakes; find out what they are and try to avoid them next time.
• If you got something like 75 you're doing fine but you did miss something significant, probably more than just a minor thing. Figure out what it was and make a plan to fix the problem for next time.
• If you got something like 55 you should be concerned. You are still in position to improve greatly and get an excellent grade at the end, but what you missed is quite significant and you are at the risk of finding yourself far behind. You must analyze what happened - perhaps it was a minor mishap, but more likely you misunderstood something major or something major is missing in your background. Find out what it is and try to come up with a realistic strategy to overcome the difficulty!
• If you got something like 35, most likely you are not gaining much from this class and you should consider dropping it, unless you are convinced that you fully understand the cause of your difficulty (you were very sick, you really couldn't study at all for the two weeks before the exam because of some unusual circumstances, something like that) and you feel confident you have a fix for next time. The deadline for dropping a class this semester is soon: Sunday March 9.

#### Appeals.

Remember! Grading is a difficult process and mistakes always happen - solutions get misread, parts are forgotten, grades are not added up correctly. You must read your exam and make sure that you understand how it was graded. If you disagree with anything, don't hesitate to complain! Dror graded everything, so appeals should go directly to him.

The deadline to start the appeal process is Wednesday March 19 at class time.

## Solution Set

Solutions by Tout can be found here